Technically I believe we are a Democratic Republic, as we directly vote on laws within our states. Legal Marijuana, for instance, was chosen by the Citizens of my state, Oregon; not by elected politicians.
that varies from state to state. California, for example, does have direct ballot initiatives at the state level. New York State (where I live) generally does not, though they aren't excluded by the state constitution, there is just not a very strong tradition of direct ballot measures here.
every state in the U.S. implements representative democracy in a slightly different way.
https://en.wikipedia.org/wiki/Federal_republic
The citizens of Oregon do get to vote on Initiatives and Referendums, that is something which is relatively new to the state 1908, and not the way states outside of the Western part of the country operate.
https://en.wikipedia.org/wiki/Direct_democracy_in_Oregon ^^ This page has a nice map showing other states as well.
In many of the blue states on the map constitutional amendments for the state can be proposed and voted on but the majority of laws are simply written by the legislature.
A federal republic of democracies at least?